Utility and Choice

(Largely from NS Chapter 2)

Motivation

Consider a decision you recently made?


  • Define this decision clearly.

  • How do you think you decided among these options?

2 minutes: discuss with your neighbour

Suppose I asked you

‘State a rule that governs (determines/characterizes) how people do make decisions’…


I want this rule to be…

  1. Informative (it rules out at least some sets of choices)

  2. Predictive (people rarely if ever violate this rule)

Similar question:

‘State a rule that governs how people should make decisions’..


By ‘should’ I mean that they will not regret having made decisions in this way.

2 minutes: discuss with your neighbour

If people did follow these rules, what would this imply and predict?

Rules defined as ‘axioms about preferences’

‘Standard axioms’ \(\rightarrow\) (imply that) choices can be expressed by ‘individuals maximising utility functions subject to their budget constraints

\(\rightarrow\) yields predictions for individual behavior, markets, etc.

Lecture 2, Ch.2 – Utility and Choice – coverage

Learn, understand, be able to explain and explain:

  • ‘Utility’, how it’s defined

  • Key assumptions about preferences/choices; their implication

  • Depict preferences/utility with ‘indifference curves’

    • … examples of ‘perfect substitutes’ and ‘perfect complements’
  • ‘Budget constraints’, compute and model them

  • Maximising utility subject to constraints

    • optimisation condition for this

  • Depict and interpret optimisation with indifference curves and budget constraints

Utility

Utility
“The pleasure or satisfaction that people get from their economic activity.”

Alt: The thing that people maximise when making economic decisions

How is this used?


Utility will be expressed as a single number that arises from the combination of all goods and services consumed.


Essentially, economists assume that ‘when making a choice among all available and feasible options, an individual will choose the one that yields the greatest utility

Utility from two goods

\[Utility = U(X,Y; other)\]


  • Leisure and ‘goods consumption’

  • Food and non-food

  • Coffee and tea (holding all else constant)

\[Utility = U(X,Y)\]


Maths revision: \(U(X,Y)\) expresses a function with two arguments, X and Y.


\(U(X,Y)\) must take some value for every positive value of X and Y.

This expresses a general function; I haven’t specified what this function is

  • E.g., it could be \(U(X,Y)=\sqrt(XY)\)

Measuring and comparing utilities

Utility is not ‘observable and measurable in utils’

  • (Unlike midi-chlorians or thetans)


  • Utility is seen to govern an individual’s choices and thus it’s only inferred indirectly, from the choices people make

Revealed preference: if Al buys a cat instead of a dog, and a dog was cheaper, we assume Al gets more utility from a cat

Interpersonal comparisons are difficult

  • Who gets ‘more’ utility?

…Interpersonal comparisons are difficult

  • Transfer from Al to Betty: Is the reduction in Al’s utility more or less than the increase in Betty’s?

Standard assumptions about preferences (‘axioms’)

  1. Completeness
  1. Transitivity (internal consistency)
  1. More is Better (nonsatiation)

1. Completeness

\[A \succ B, B \succ A, \: or \: A \sim B \]

Fancy notation: Either A preferred to B, B preferred to A, or A indifferent to B

Forbidden: “I can’t choose between a ski holiday and a beach holiday, but I am not indifferent”

2. Transitivity

\[ A \succ B \: and \: B \succ C \rightarrow A \succ C \]

  • Similar for indifference (\(\sim\))

If I prefer an Apple to a Banana and a Banana to Cherry

then I prefer an Apple to a Cherry.


If not \(\rightarrow\) money pump.

3. More is better

(similar to nonsatiation, ‘monotonicity’)

Draw this!

If the product is a ‘bad’ (e.g., pollution), redefine as the absence of the product

Who cares?


If people obey the first two assumptions (axioms),\(^\ast\) they will make choices in a way consistent with maximising a (continuous) utility function


*(and also ‘continuity’, which you can ignore for this module)

  • How can we compare the “?” areas? Which are preferred?
  • \(\rightarrow\) Compare utilities, depict using Indifference Curves

Indifference curves

Indifference curve
A curve that shows all the combinations of goods or services that provide the same level of utility

Indifference curves

Indifference curve
A curve that shows all the combinations of goods or services that provide the same level of utility


Formally (for 2 goods), the set of pairs of \(\{X,Y\}\) such that \(U(X,Y)=c\)
for some constant \(c\)


Warning: Indifference curves help us depict utility functions; a single indifference curve is not itself a utility function!

Credit: www2.econ.iastate.edu

Credit: Frank’s Economics on the web (MIT)

Properties of indifference curves

Rank order of preference/indifference between points A-E.

Q1: How do we know \(E \succ B\) ?

Q: How do we know \(E \succ A\) ?

Why ‘voluntary trade’?

The indifference curve offers some intuition.

Marginal rate of substitution (MRS)

MRS = Absolute value of slope of indifference curve


‘Rate at which you’re willing to forgo consuming \(Y\) to consume one more \(X\)

Back to fig 2.2 (board or visualiser)


  • A to B: willing to give up 2 hamburgers to get 1 more soda.
  • \(\rightarrow\) slope \(-2\), \(MRS=2\)


  • From B to C? (think about it)
  • …willing to give up 1 hamburger to get 1 more soda \(\rightarrow MRS =1\)


  • C to D?
  • \(MRS = \frac{1}{2}\)


Note the decline: ‘diminishing MRS’: may reflect satiation

Preference for variety/balance

Fig 2.3

Indifference curve map

Indifference curves never cross! And are never upwards sloping!

Illustrating particular preferences (NS fig 2.5)

App 2.3: Product positioning in marketing (read at home, see handout)

Definitions: Perfect substitutes and complements

Perfect substitutes

Perfect substitutes

Goods A and B are Perfect Substitutes when an individual’s utility is linear in these goods

when she is always willing to trade off A for B at a fixed rate (not necessarily 1 for 1)

Perfect complements

Goods A and B are Perfect Complements when an individual only gains utility from (more) A if she also consumes a defined (additional) amount of B, and vice-versa


These goods are ‘enjoyed only in fixed proportions’.

Choices are subject to constraints :(


You cannot spend more than your (lifetime) income/wealth


\(\rightarrow\) budget constraint.

Budget constraint for two goods, slope $-P_x/P_y$

Budget constraint for two goods, slope \(-P_x/P_y\)

Budget constraint algebra

If I spend all my income (I will do over a ‘relevant lifetime’):

Expenditure on X + Expenditure on Y = Income (I)

\[P_X X + P_Y Y = I \]

To see how \(Y\) trades off against \(X\), rearrange this to:

\[Y = -\frac{P_X}{P_Y} X + \frac{I}{P_Y}\]

Expenditure on X + Expenditure on Y = Income (I)

\[P_X X + P_Y Y = I \]

\[Y = -\frac{P_X}{P_Y} X + \frac{I}{P_Y}\]


Intercept \(\frac{I}{P_Y}\): amount of Y you can buy if you only buy Y

Slope \(-\frac{P_X}{P_Y}\): how much Y you must give up to get another X

Utility maximization


If slope budget constraint = slope indifc curve at point X,Y \(\rightarrow\) \[P_X/P_Y = MRS(X,Y)\]

  • Warning: This equality holds at an optimal choice; it doesn’t hold everywhere.

At an optimal consumption choice (given above assumptions)


  • Consume all of income; locate on budget line; follows from ‘more is better’


  • Psychic tradeoff (MRS) equals market tradeoff (\(P_X/P_Y\)) if consuming both goods

Psychic tradeoff (MRS) equals market tradeoff (\(P_X/P_Y\)) if consuming both goods



Key intuition:

  • If I can give up X for (buy less X, get more Y) at some rate, and the benefit I get from doing this is at a different rate,


  • …then I can make myself better off.


  • Thus the original point could not have been optimal.

More insight (mathy: ignore if this freaks you out)

Recall \(U=U(X,Y)\).

\[U_X(X,Y) := MU_X(X,Y)\]

Derivative w/ respect to X: rate utility increases if we add a little X, holding Y constant

… Similarly for \(MU_Y\).


  • MRS: ‘how much Y would I be willing to give up to get a unit of X’?


  • Ans: Depends on marginal benefit of each … we can show \(MRS(X,Y)=\frac{MU_{X}}{MU_{Y}}\)

Rearranging the utility maximising condition yields more intuition:

\[P_X/P_Y = MRS = MU_X/MU_Y\]

(at each consumption point X,Y)


  • \[\frac{MU_X}{P_X} = \frac{MU_Y}{P_Y}\]


  • Same ‘bang for each buck’ (if optimising)

Caveat on ‘corner solutions’

  • If you are consuming both goods and optimising, \(P_X/P_Y = MRS = MU_X/MU_Y\) must hold


Advanced: This is a ‘necessary but not sufficient condition’, sufficient if DMRS everywhere



But…

But…

  • If you are consuming both goods and optimising, \(P_X/P_Y = MRS = MU_X/MU_Y\) must hold


  • But you might consume none of some good (say X):


  • if even with no X, \(MU_X/P_X<MU_Y/P_Y\), the marginal utility of the first unit ‘per pound’ is lower

App 2.4: ticket scalping

App 2.5: What’s a rich uncle’s promise worth?


Using the model of choice

  1. Why do people spend their money on different things?
  2. What do different preferences/indifference curves imply for choices?

Move to ppt slides here beginning with ‘Utility Maximization: A Graphical View’

Algebraic/numerical examples

Consider: are these ‘perfect substitutes’ for someone who wants caffeine, but has no taste buds?

Perfect substitutes, but not identical, e.g.,

\[U(X,Y)=4X+3Y\]

Rates each increase utility per-unit (derivative) are constant: \(MU_X = 4\), \(MU_Y = 3\)

So (for perfect substitutes) buy the one that increases it more *per-

With perfect substitutes: ‘Bang for the buck’ rule

\[U(X,Y)=4X+3Y\]


  • Compare \(MU_X/P_X\) to \(MU_Y/P_Y\)
  • Here, if \(4/P_X > 3/P_Y\), then buy X

  • if \(4/P_X < 3/P_Y\), then buy Y; (if equal, buy either)

  • Rearranging, if \(P_X < 4/3 P_Y\), buy X … etc.

Warning: If not perfect substitutes, MU ratios depend on consumption levels.

Perfect complements

Mathematical function example:

\[U(X,Y)=min(2X,Y)\]

E.g., X: bicycle frames, Y: wheels.


Warning: this min function looks backwards, but it’s correct; see notes

  • Shortcut: figure out the proportions it will be consumed in
    • determine cost of ‘1 bundle of the combo’ at given prices
    • … then buy as many such bundles as you can afford

Middle-ground (*)

A Cobb-Douglas example

\[ U(X,Y)=\sqrt(XY) \]

\[MU_X = \frac{\partial}{\partial X} (XY)^{1/2} = \frac{1}{2} (Y/X)^{1/2}\]

\[MU_Y = \frac{1}{2} (X/Y)^{1/2}\]


Here, amount of Y you’d give up to get a unit of X:

\[MRS(X,Y)= MU_X/MU_Y = Y/X\]

{Check reasonable: The more Y I’ve , the more Y I’d give up to get another X :)}

…Cobb-Douglas ctd

\[MRS(X,Y)= MU_X/MU_Y = Y/X\]

Here utility-maximization requires, at optimal choices of X and Y: \[MRS(X,Y)= Y/X = P_X/P_Y\]

For any price ratio, find ratio of Y & X.

With prices and income, \(I\), find consumption of X & Y.

Rearranging optimization condition:

\[Y P_Y = X P_X\]

Optimization condition for this particular utility function:

\[Y P_Y = X P_X\]

Combining this with the budget constraint \[P_X X + P_Y Y = I\]

solve for X & Y, as fncns of prices & income (see notes) \(\rightarrow\)

\[Y = I/(2P_Y)\] \[X = I/(2P_X)\]

Second problem set: covers NS chapter 2 (and chapter 3) – see bottom

Preferences, Utility, Consumer optimization,


individual and market demand curves.

This is a very large problem set (others are smaller).